reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
for n be Nat, f be PartFunc of S,T st 1 <= n & f is_differentiable_on n,Z
holds
 for i be Nat st i <= n holds diff(-f,i,Z) = -diff(f,i,Z)
proof
   let n be Nat, f be PartFunc of S,T;
   assume A1: 1 <= n & f is_differentiable_on n,Z;
   let i be Nat;
   assume i <= n; then
   diff((-1)(#)f,i,Z) = (-1)(#)diff(f,i,Z) by Th24,A1; then
   diff((-1)(#)f,Z).i = -diff(f,i,Z) by VFUNCT_1:23;
   hence diff(-f,i,Z) = -diff(f,i,Z) by VFUNCT_1:23;
end;
